v2009.01.01 - Convex Optimization

E.10. ALTERNATING PROJECTION 693
(a feasible point) whose existence is guaranteed by virtue of the fact that each
and every point in the convex intersection is in one-to-one correspondence
with fixed points of the nonexpansive projection product.
Bauschke & Borwein [27,2] argue that any sequence monotonic in the
sense of Fejér is convergent. E.20
E.10.2.0.1 Definition. Fejér monotonicity. [233]
Given closed convex set C ≠ ∅ , then a sequence x i ∈ R n , i=0, 1, 2..., is
monotonic in the sense of Fejér with respect to C iff
‖x i+1 − c‖ ≤ ‖x i − c‖ for all i≥0 and each and every c ∈ C (1934)
Given x 0 ∆ = b , if we express each iteration of alternating projection by
x i+1 = P 1 P 2 x i , i=0, 1, 2... (1935)
and define any fixed point a =P 1 P 2 a , then sequence x i is Fejér monotone
with respect to fixed point a because
‖P 1 P 2 x i − a‖ ≤ ‖x i − a‖ ∀i ≥ 0 (1936)
by nonexpansivity. The nonincreasing sequence ‖P 1 P 2 x i − a‖ is bounded
below hence convergent because any bounded monotonic sequence in R
is convergent; [222,1.2] [37,1.1] P 1 P 2 x i+1 = P 1 P 2 x i = x i+1 . Sequence
x i therefore converges to some fixed point. If the intersection C 1 ∩ C 2
is nonempty, convergence is to some point there by the distance theorem.
Otherwise, x i converges to a point in C 1 of minimum distance to C 2 .
E.10.2.0.2 Example. Hyperplane/orthant intersection.
Find a feasible point (1929) belonging to the nonempty intersection of two
convex sets: given A∈ R m×n , β ∈ R(A)
C 1 ∩ C 2 = R n + ∩ A = {y | y ≽ 0} ∩ {y | Ay = β} ⊂ R n (1937)
the nonnegative orthant with affine subset A an intersection of hyperplanes.
Projection of an iterate x i ∈ R n on A is calculated
P 2 x i = x i − A T (AA T ) −1 (Ax i − β) (1828)
E.20 Other authors prove convergence by different means; e.g., [149] [55].
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